The two popular methods of testing bias in word embeddings are word analogy tests and word association tests.
In word analogy tests, given two words in a certain syntactic or semantic relation (man \(\rightarrow \)  king), the goal is generate a word that is in similar relation to a given word (woman \(\rightarrow \)  queen).
[1]} showed that word embeddings capture syntactic and semantic word analogies, e.g., gender, morphology etc.
[2]} build on this observation to study gender bias.
They show that word embeddings capture several undesired gender biases (semantic relations) e.g.
doctor : man :: woman : nurse.
[3]} extend this to show that word embeddings capture several stereotypical biases such as racial and religious biases.
